In most new build estates there are similar restrictions in the missives - usually things on pets, not keeping vans on driveways, not changing the colour of your front door etc. These are mainly there to ensure that the estate is orderly and peaceful while the builder is still selling property there. In practice however, most people do what they want after that initial period. Unless someone was to complain to the builders, lawyers or factors, there would not be an issue. Whilst technically you would be in breach of your missives, if you are not causing a problem, you would be very unlucky if someone was to complain and I am not sure how seriously they would take it and whether they wanted to risk taking you to court to remove the dogs from the property.
If I were you, I would ask your lawyer for his opinion - after all you are paying him - but in the end, I would go ahead and just be a good neighbour and enjoy your new home. It's not like you are breeding pit bulls in the back garden is it?